Critical Asymmetry for Giant Diffusion of Active Brownian Particles

We study the effect of an asymmetry on the transport properties of an active Brownian particle. We demonstrate the existence of a critical force or, more generally, of a critical asymmetry that separates parameter regimes of giant diffusion from those with reliable directed transport. We derive a condition for the critical asymmetry by means of an exact expression for the diffusion coefficient and by a simplified discrete picture. A critical asymmetry, as predicted by the simple model, is also found in a detailed model of coupled molecular motors displaying bidirectional motion.

Figure 1

The velocity potential (a) and sample trajectories (b) of an ABP in the symmetric (dotted) and in the asymmetric case (solid, $F=0.085$), respectively. In (b): simulations of Eq. (5) with $Q=0.05$, $F=0$ (dotted) and $F=0.02$ (solid).

Figure 2

Transport properties of active Brownian particles with an asymmetry. Mean velocity (a) and diffusion coefficient (b) vs bias for different noise levels $Q$. Exact result Eq. (4) compared to numerical simulations of Eq. (5) (symbols) and to the two-state theory Eq. (6) (dotted lines). Forces within the shaded region ($F\in (-F_{\mathrm{crit}},F_{\mathrm{crit}})$ with $F_{\mathrm{crit}}\approx 0.0852$) lead to giant diffusion in the weak noise limit; outside this window, reliable transport is observed. Panels (c)–(g) show the velocity potentials at the forces as indicated.

Figure 3

Coupled molecular motors. Following Ref. 6 and working in nondimensional units, we use a rate $r_1(x)$ which is $r_1=500$ in a region of width $d=0.2$ centered at the potential minima, otherwise $r_1(x)=0$. The rate $r_0=40$ is spatially independent. Other parameters: $W=1$, $L=1$, $\lambda =0.01$, simulation time step $\Delta t={10}^{-3}$; $q$ is incommensurate with $L$.

Figure 4

Transport properties of an assembly of molecular motors. Mean velocity (a) and diffusion coefficient (b) for a symmetric spatial potential ($a=L/2$) vs external force for different numbers of motors as indicated. The inset shows the ratio $R_P$ Eq. (8) vs the number of motors for three values of $F$ indicated by arrows in (b). Panels (c) and (d) show the mean velocity and effective diffusion coefficient for an asymmetric potential ($a=0.4L$); note that the range of forces has shifted to negative values. Insets in (d): ratios $R_P$ measured at six values of the force close to the critical values.